29.10. EXERCISES 579

Show that this is the same as finding y1, · · · ,yn where

yk+1− yk

t/n= yk

In place of y′ (s) = y(s) , you have yk+1−ykt/n = yk. Now show that yn =

(1+ t

n

)n y0.

What is the limit as n→ ∞?

83. Suppose on an interval [a,a+h] , you have y′ (t) = f (t,y(t)) and z(t) = y(a) +(t−a) f (a,y(a)) . Suppose also the solution y has bounded continuous second deriva-tives. Show that |y(h)− z(h)| < Ch2 for some constant C. You will need to useTaylor’s theorem. This is the local error for the Euler method.